SAS/ETS Software

Time Series Analysis

Panel Data Regression Analysis

The PANEL procedure analyzes a class of linear econometric models that commonly arise when time series and cross-sectional data are combined. The PANEL procedure analyzes panel
data sets that consist of multiple time series observations on each of several individuals or cross-sectional units. The performance of any estimation procedure for the model
regression parameters depends on the statistical characteristics of the error components in the model. The PANEL procedure estimates the regression parameters in the preceding
model under several common error structures, including one and two-way fixed and random effects.

Regression with Autocorrelated and Heteroscedastic Errors

In regression analysis, if the error terms are not independent (autocorrelated), the efficiency of the ordinary
least-square (OLS) parameter estimates is adversely affected and the standard error estimates are biased. This happens
frequently with time series data.

Ordinary regression analysis assumes that the error variance is the same for all observations. When the error variance is
not constant, the data are said to be heteroscedastic, and ordinary least-squares estimates are inefficient.

The AUTOREG procedure estimates and forecasts linear regression models for time series data when the errors are
autocorrelated or heteroscedastic. The autoregressive error model is used to correct for autocorrelation, and the
generalized autoregressive conditional heteroscedasticity (GARCH) model and its variants are used to model and correct for
heteroscedasticity.

The AUTOREG procedure supports the following variations of the GARCH model:

generalized ARCH (GARCH)

integrated GARCH (IGARCH)

exponential GARCH (EGARCH)

GARCH-in-mean (GARCH-M)

The procedure can also analyze models that combine autoregressive errors and GARCH-type heteroscedasticity. The maximum likelihood method is used for GARCH models and for mixed AR-GARCH models.
Four estimation methods are supported for the autoregressive error model:

ARIMA (Box-Jenkins) and ARIMAX (Box-Tiao) Modeling and Forecasting

The ARIMA procedure analyzes and forecasts equally spaced univariate time series data, transfer function data, and
intervention data using the autoregressive moving-average (ARMA) model or the more general autoregressive integrated
moving-average (ARIMA) model. An ARIMA model predicts a value in a response time series as a linear combination of its own
past values, past errors, and current and past values of other time series.

The ARIMA procedure provides a comprehensive set of tools for univariate time series model identification, parameter
estimation, and forecasting. It offers great flexibility in the kinds of ARIMA or ARIMAX models that can be analyzed. The
procedure supports seasonal, subset, and factored ARIMA models; intervention or interrupted time series models; multiple
regression analysis with ARIMA errors; and transfer function models of any complexity.

Polynomial Distributed Lag Regression

The PDLREG procedure estimates regression models for time series data in which the effects of some of the regressor
variables are distributed across time. The distributed lag model assumes that the effect of an independent variable, X, on a
dependent variable, Y, is distributed over time. If the value of X at time t changes, Y experiences some immediate
effect at time t, and it also experiences delayed effects at times t + 1, t + 2, and so on up to time
t + p, for some limit p. The distribution of the lagged effects is modeled by Almon lag polynomials.
The coefficients of the lagged values of the regressor are assumed to lie on a polynomial curve.

Regression models supported by PROC PDLREG can include any number of regressors with distribution lags and any number of
covariates (simple regressors without lag distributions).

You can specify a minimum degree and a maximum degree for the lag distribution polynomial, and the procedure fits
polynomials for all degrees in the specified range.

The PDLREG procedure can also test for autocorrelated residuals and perform autocorrelated error correction using the
autoregressive error model. You can specify any order autoregressive error model and several different estimation methods
for the autoregressive model, including exact maximum likelihood.

State Space Modeling and Forecasting

The SSM procedure is useful for automatic modeling and forecasting of several interrelated time series with or without a feedback relationship. The procedure analyzes and forecasts
multivariate time series using the state space model. It is appropriate for jointly forecasting several related time series that have dynamic interactions. By taking into account
the autocorrelations among the whole set of variables, the SSM procedure may give better forecasts than methods that model each series separately. By default, the SSM procedure
automatically selects a state space model appropriate for the time series, making the procedure a good tool for automatic forecasting of multivariate time series.

Spectral Analysis

Spectral analysis is a statistical approach to detecting regular cyclical patterns, or periodicities, in transformed time
series data.

The SPECTRA procedure produces estimates of the spectral and cross-spectral densities of a multivariate time series.
Estimates of the spectral and cross-spectral densities of a multivariate time series are produced using a finite Fourier
transform to obtain periodograms and cross-periodograms. The periodogram ordinates are smoothed by a moving average to
produce estimated spectral and cross-spectral densities. PROC SPECTRA can also test whether the data are white noise.

The SPECTRA procedure performs spectral and cross-spectral analysis of time series